Let $a,b,c$ be positive real numbers for which $a+b+c=3$. Prove the inequality \[\frac{a^3+2}{b+2}+\frac{b^3+2}{c+2}+\frac{c^3+2}{a+2}\ge3\]

If $ a$, $ b$, $ c$ are the sidelengths of a triangle, then prove that $ \frac {3\left(a^4 \plus{} b^4 \plus{} c^4\right)}{\left(a^2 \plus{} b^2 \plus{} c^2\right)^2} \plus{} \frac {bc \plus{} ca \plus{} ab}{a^2 \plus{} b^2 \plus{} c^2}\geq 2$.

A square with center $F$ was constructed on the side $AC$ of triangle $ABC$ outside it. After this, everything was erased except $F$ and the midpoints $N,K$ of sides $BC,AB$. Restore the triangle.

Let $n > 1$ be an integer and let $a_1, a_2,... , a_n$ be $n$ different integers. Show that the polynomial $f(x) = (x -a_1)(x - a_2)\cdot ... \cdot (x -a_n) - 1$ is not divisible by any polynomial with integer coefficients and of degree greater than zero but less than $n$ and such that the highest power of $x$ has coefficient $1$.

In the plane, let $C$ be a closed convex set that contains $(0,0)$ but no other point with integer coordinates. Suppose that $A(C)$, the area of $C$, is equally distributed among the four quadrants. Prove that $A(C) \leq 4.$

Two players are playing a game with $n$ pieces. At each turn, the player takes $2^i$ pieces where $i \geq 0$. The player who takes the last piece will win the game. If the game is played for each $n=1000, 2000, 2011, 3000, 4000$ once, in how many of them the first player can guarantee to win? $\textbf{(A)}\ 4 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ \text{None}$

In convex quadrilateral $ABCD$, $\angle CAB = \angle BCD$. $P$ lies on line $BC$ such that $AP = PC$, $Q$ lies on line $AP$ such that $AC$ and $DQ$ are parallel, $R$ is the point of intersection of lines $AB$ and $CD$, and $S$ is the point of intersection of lines $AC$ and $QR$. Line $AD$ meets the circumcircle of $AQS$ again at $T$. Prove that $AB$ and $QT$ are parallel.

ABC  is a triangle with coordinates A =(2, 6), B =(0, 0), and C =(14, 0). (a) Let P  be the midpoint of AB. Determine the equation of the line perpendicular to AB passing through P. (b) Let Q be the point on line BC  for which PQ is perpendicular to AB. Determine the length of AQ. (c) There is a (unique) circle passing through the points A, B, and C. Determine the radius of this circle.

Let $a,b,c$ be real numbers such that $0 \leq a \leq b \leq c$ and $a+b+c=ab+bc+ca >0.$ Prove that $\sqrt{bc}(a+1) \geq 2$ and determine the equality cases. (Edit: Proposed by sir Leonard Giugiuc, Romania)

A triangle has altitudes of length $15$, $21$, and $35$. Find its area.

Let $a, b, c, d$ be integers such that $a > b > c > d \ge -2021$ and $$\frac{a + b}{b + c}=\frac{c + d}{d + a}$$ (and $b + c \ne 0 \ne d + a$). What is the maximum possible value of $ac$?

The inscribed circle of the triangle $ABC{}$ touches the sides $BC,CA$ and $AB{}$ at $A',B'$ and $C'{}$ respectively. Let $I_a$ be the $A$-excenter of $ABC{}.$ Prove that $I_aA'$ is perpendicular to the line determined by the circumcenters of $I_aBC'$ and $I_aCB'.$

Let $ABC$ be an acute triangle with the circumcircle $k$ and incenter $I$. The perpendicular through $I$ in $CI$ intersects segment $[BC]$ in $U$ and $k$ in $V$. In particular $V$ and $A$ are on different sides of $BC$. The parallel line through $U$ to $AI$ intersects $AV$ in $X$. Prove: If $XI$ and $AI$ are perpendicular to each other, then $XI$ intersects segment $[AC]$ in its midpoint $M$. [i](Notation: $[\cdot]$ denotes the line segment.)[/i]

Let $n$ be a positive integer. A finite set of integers is called $n$-divided if there are exactly $n$ ways to partition this set into two subsets with equal sums. For example, the set $\{1, 3, 4, 5, 6, 7\}$ is $2$-divided because the only ways to partition it into two subsets with equal sums is by dividing it into $\{1, 3, 4, 5\}$ and $\{6, 7\}$, or $\{1, 5, 7\}$ and $\{3, 4, 6\}$. Find all the integers $n > 0$ for which there exists a $n$-divided set. Proposed by [i]Martin Rakovsky, France[/i]

Let $k$ be a fixed natural number. In the infinite number of real line, each integer is colored with color ..., red, green, blue, red, green, blue, ... and so on. A number of flea settles at first at integer points. On each turn, a flea will jump over the other tick so that the distance $k$ is the original distance. Formally, we may choose $2$ tails $A, B$ that are spaced $n$ and move $A$ to the different side of $B$ so the current distance is $kn$. Some fleas may occupy the same point because we consider the size of fleas very small. Determine all the values of $k$ so that, whatever the initial position of the ticks, we always get a position where all ticks land on the same color.

The vertices of the regular $n$-gon are marked with some colours (each vertex -- with one colour) in such a way, that the vertices of one colour represent the right polygon. Prove that there are two equal ones among the smaller polygons.

While doing her homework for a Momentum Learning class, Valencia draws two intersecting segments $AB = 10$ and $CD = 7$ on a plane. Across all possible configurations of those two segments, determine the maximum possible area of quadrilateral $ACBD$.

Find maximum value of number $a$ such that for any arrangement of numbers $1,2,\ldots ,10$ on a circle, we can find three consecutive numbers such their sum bigger or equal than $a$.

There are $10001$ students at an university. Some students join together to form several clubs (a student may belong to different clubs). Some clubs join together to form several societies (a club may belong to different societies). There are a total of $k$ societies. Suppose that the following conditions hold: [i]i.)[/i] Each pair of students are in exactly one club. [i]ii.)[/i] For each student and each society, the student is in exactly one club of the society. [i]iii.)[/i] Each club has an odd number of students. In addition, a club with ${2m+1}$ students ($m$ is a positive integer) is in exactly $m$ societies. Find all possible values of $k$. [i]Proposed by Guihua Gong, Puerto Rico[/i]

A number $N$ has $2009$ positive factors. What is the maximum number of positive factors that $N^2$ could have?

If the angles $ a,b,c,d,$ and $ e$ are known, what does the angle $ u$ equal? [img]http://i250.photobucket.com/albums/gg265/geometry101/GeometryImage1.jpg[/img] A. $ 108^\circ$ B. $ 2a$ C. $ 3a$ D. $ c\plus{}d$ E. $ a\plus{}c\plus{}d$

Let $ABTCD$ be a convex pentagon with area $22$ such that $AB = CD$ and the circumcircles of triangles $TAB$ and $TCD$ are internally tangent. Given that $\angle{ATD} = 90^{\circ}, \angle{BTC} = 120^{\circ}, BT = 4,$ and $CT = 5$, compute the area of triangle $TAD$.

Two perpendicular chords of a circle, $AM, BN$ , which intersect at point $K$, define on the circle four arcs with pairwise different length, with $AB$ being the smallest of them. We draw the chords $AD, BC$ with $AD // BC$ and $C, D$ different from $N, M$ . If $L$ is the intersection point of $DN, M C$ and $T$ the intersection point of $DC, KL,$ prove that $\angle KTC = \angle KNL$.

Determine all functions $f:\mathbb R\to\mathbb R$ that satisfy equation: $$ f(x^3+y^3) =f(x^3) + 3x^2f(x)f(y) + 3f(x)f(y)^2 + y^6f(y) $$ for all reals $x,y$

Nick is a runner, and his goal is to complete four laps around a circuit at an average speed of $10$ mph. If he completes the first three laps at a constant speed of only $9$ mph, what speed does he need to maintain in miles per hour on the fourth lap to achieve his goal?